Reconstructing a three-color scenery by observing it along a simple random walk path

Let {xi(n)}(n is an element of Z) be a three-color random scenery, i.e., a random coloration of Z in three colors, such that the colors of the different points in Z are i.i.d. Let {S(n)}(n is an element of N) be a symmetric random walk starting at 0. Our main result shows that a.s., xi circle S (the composition of xi and S) determines xi up to translation and reflection. In other words, by observing the scenery along the random walk path S, we can a.s. reconstruct xi up to translation and reflection. This result allows us to give a positive answer to the question of PI. Kesten of whether one can a.s. detect a single defect in a three-color random scenery by observing it only along a random walk path. (C) 1999 John Wiley & Sons, Inc.